Research

Ion electrokinetics in nonuniformly charged capillaries

Electroosmosis, the transport of electrolytes with the application of electric fields, is a linear effect (the fluid velocity varies linearly with respect the electric field amplitude) and has applications in energy and engineering processes such as mixing. 


In our work, we show that traveling wave charges on electrodes, give rise to a zero (or Goldstone) mode in the electrolyte velocity. This is a nonlinear effect caused by continuous symmetry-breaking. The fluid velocity here varies quadratically with respect to the electric field generated by the charge distribution and does not vanish when averaged over the period of oscillation of the field. As the effect becomes more pronounced by reducing the characteristic lengthscale of the system, it can be employed for unidirectional transport of electrolytes in small capillaries. 

Preprint

Wall and body modes in rigidly-rotating odd viscous liquids

Non-axisymmetric three-dimensional incompressible or two-dimensional compressible odd viscous liquids, rotating rigidly give rise to both oscillatory and evanescent inertial-like waves or a combination thereof (which we call of mixed type). 

These waves precess in a prograde or retrograde manner with respect to the rotating frame and become prominent close to a solid boundary. The oscillatory and evanescent waves resemble, respectively, the body and wall modes observed in (non-odd) rotating Rayleigh-B\'enard convection . We show that the three types of waves (wall, body or mixed) can be classified with respect to pairs of planar wavenumbers which are complex, real or a combination, respectively. Experimentally, by observing the precession rate of the patterns, it would be possible to determine the largely unknown values of the odd viscosity coefficients. 

This formulation recovers as special cases recent studies of equatorial or topological waves in two dimensional odd viscous liquids which provided examples of the bulk-interface correspondence.

Preprint

Odd viscosity-induced migration of thermocapillary droplets

A droplet of a classical liquid surrounded by a cold gas placed on a hot substrate is accompanied by unremitting internal circulations, while the droplet remains immobile. Two identical cells with opposite sense of circulation form in the interior due to the thermocapillary effect induced by the gas and substrate temperature difference (this is described in the left column of the adjacent figures, generated by Aaveg Aggarwal using Comsol and is a well-known effect, see Alexander Oron, Stephen H. Davis, and S. George Bankoff Rev. Mod. Phys. 69, 931, 1997). 


We show that under the same conditions, a droplet composed of an odd viscous liquid exerts a compressive stress on the cell rotating in one sense and tensile on the cell rotating in the opposite sense resulting in a tilted droplet configuration. A sufficiently strong thermal gradient leads the contact angles to overcome hysteresis effects and induces droplet migration.


The first cartoon above explains the physical mechanism. The last three rows show a Comsol-generated snapshot of a three-dimensional migrating thermocapillary droplet (right column, non-zero odd viscosity coefficient). 

Publication

31. "Thermocapillary Migrating Odd Viscous Droplets" Physical Review Letters 131, 198201 (2023) A. Aggarwal, E. Kirkinis, & M. Olvera de la Cruz 

Chiral and achiral propulsion and separation in liquids

Industrial and laboratory processes give rise to dissimilar units which can prove useful only after segregation. In the context of cell biology these units can be proteins, organelles and macromolecules. Separation can proceed by taking advantage of the size of the particles, their chemical properties or their shape. In this work we develop separation mechanisms that only depend on  particle shape and can be employed with a minimal degree of sophistication and expense in both particle preparation and system calibration Chiral particles can also be propelled in a base liquid with rotational degrees of freedom. These effects will also be investigated for suspensions of particles of completely random shape.  

Publications

32. 'Hydrodynamics of thermally-driven chiral suspensions' Journal of Fluid Mechanics, 977: A8, (2023) E.Kirkinis, A.V.Andreev & M. Olvera de la Cruz

28. 'Activity-induced propulsion and separation of passive chiral particles in liquids' Physical Review Fluids, 8, 023302 (2023), E.Kirkinis & M. Olvera de la Cruz

Figure above: Cross flow of a Newtonian liquid with a thermal gradient gives rise to chiral structure propulsion and separation according to their handedness. In turn, the chiral suspension alters the liquid flow which thus acquires a transverse (chiral) velocity component. Since observation of the predicted effects requires a low degree of sophistication, our work provides an efficient and inexpensive approach to test and calibrate chiral particle propulsion and separation strategies.

Preprint

arXiv:2306.16509


Loss of convexity in rotating odd viscous liquids

A non-rotating odd viscous liquid can give rise to Taylor columns, that is a column of liquid circumscribing a body moving slowly along the axis of a cylinder (for instance) as in the left figure. In the presence of rotation however, the dispersion relation may lose convexity and lead to unconventional fluid-flow behavior such as soliton tails and dispersive shocks. Here on the right we display the loss of equlibrium due to rotation and transition to a time-dependent state.

Preprint

Inertial-like waves in rigidly-rotating odd viscous liquids E.Kirkinis & M. Olvera de la Cruz arXiv:2307.00415

See also: 'Taylor columns and inertial-like waves in a three-dimensional odd viscous liquid' Journal of Fluid Mechanics, 973: A30 , (2023) E.Kirkinis & M. Olvera de la Cruz

Microscale Interfacial Flows and Active Matter

With Stephen H. Davis at Northwestern and Anton V. Andreev in Washington we developed a number of mechanisms (odd viscosity, viscous heating, magnetic torque, surface shearing) that stabilize an unstable thin liquid film and derived the corresponding bifurcation criteria and amplitude equations. Below I outline the magnetic torque case:

The rotational degrees of freedom endowed on a ferromagnetic liquid by its internal structure (Dahler & Scriven 1961, Shliomis 1967) may lead to the suppression of van der Waals-driven film rupture. A magnetic torque can drive the suspended particles in such a way that at the liquid-gas interface their collective rotation provides a dominant horizontal component for the fluid velocity which generates a finite-amplitude traveling wave which opposes rupture.  

The presence of non-zero value of magnetic torque N breaks the O(2) reflection symmetry of the liquid-gas interface evolution equation and endows these systems with an SO(2) symmetry such that solutions become invariant with respect to parallel translations only. 

This broken symmetry induces patterns that slowly drift in the frame of reference moving with velocity N. These patterns are nonlinear viscous interfacial waves which are stabilizing, for instance against the van der Waals-induced instability for certain values of N.

Publications

26. 'Activity-induced migration of magnetic droplets' Journal of Fluid Mechanics, 955: A10 (2023), A. Aggarwal, E. Kirkinis & M. Olvera de la Cruz

23. ‘Healing of thermocapillary film rupture by viscous heating’ Journal of Fluid Mechanics 872, 308-326 (2019) E.Kirkinis & A.V.Andreev

22. ‘Magnetic torque-induced suppression of van der Waals-driven thin liquid film rupture’ Journal of Fluid Mechanics, 813, 991-1006 (2017) E.Kirkinis

21. ‘Stabilization mechanisms in the evolution of thin liquid-films’ Proceedings of the Royal Society of London A, 20150651, 471 (2015) E.Kirkinis & S.H.Davis

Taylor halos and Taylor spears

Recently, we showed that three-dimensional  odd-viscous liquids  give rise to inertial waves and Taylor columns. This is so because odd viscous liquids are endowed with an intrinsic mechanism that tends to restore a displaced particle back to its original position, a certain type of "elasticity". 

Data can propagate obliquely to the center axis, along a Monge cone forming Taylor spears as in the adjacent figure. Characteristics that are parallel to the center axis are responsible for the generation of the commonly occurring Taylor columns.

Publications

29. 'Taylor halos and Taylor spears in odd viscous liquids' Physics of Fluids 35, 101702 (2023) E.Kirkinis & M. Olvera de la Cruz     

Contact lines, wetting and spreading

Physical processes such as the spreading of adhesives, solid surface coating by a liquid are all characterized by a sharp interface separating a liquid from a gas phase.  The intersection of this interface with a solid substrate defines the contact line. This work has important applications to ink-jet printing and medicine, for instance in the dynamics of evaporation of the tear film at the cornea.

With Stephen H. Davis at Northwestern we developed a new theory - low Reynolds number hydrodynamics - of liquid slippage on a solid substrate near a moving contact line consistent with experimental observations.  We theoretical predicted liquid behavior, yet to be seen in experiment for example the occurrence of a cascade of eddies  (moving Moffatt vortices, Moffatt1964) in the vicinity of the contact line.

Highlighted by Northwestern Engineering News: New Fluid-Dynamic Slip Law Subsumes 40 Years of Research Findings could lead to improved manufacturing, medical processes Jun 10, 2013 Article by Sarah Ostman

Steve Davis Public Lecture : A History of Moving Contact Lines

Publications

20. ‘Moffatt vortices induced by the motion of a contact line’ Journal of Fluid Mechanics 746, (2014) R3 E.Kirkinis & S.H. Davis 

19. ‘Hydrodynamic theory of liquid slippage on a solid substrate near a moving contact line’ Physical Review Letters, 234503 110 (2013) E.Kirkinis & S.H. Davis 

Hydrodynamics of magnetic nanoparticles in living systems

New and unconventional strategies are required to address crises in biological and living systems. For instance, it is expected that by 2040, the number of new cancer cases per year will rise to 29.5 million and the number of cancer-related deaths to 16.4 million in the USA alone (NIH data). In this project we will integrate three approaches, inspired by hydrodynamics and soft matter, and apply them to living systems: 

Direction I: Biological Fluids. Viscous heating. In line with recent experiments (Huang 2010) we will show that the correct way for transforming magnetic energy into therapeutic heat is to consider hydrodynamic models of magnetic particle actuation that incorporate rotational degrees of freedom (Kirkinis 2017), replacing the currently inadequate diffusive models. 


Direction II: Soft Matter. Stress alleviation. We will invent new deformation modes that would reopen compressed blood vessels and constitutive laws that will incorporate the inhomogeneous environment characterizing a tumor, employing multiscale modeling conforming to related experiments. 


Direction III: Complex Fluids. Magnetic locomotion. We will show how droplets under a magnetic field can climb a barrier working against pressure, thermal or chemical gradients and gravity; move on the underside of a plate; climb and get past obstacles and even deform to enter narrow passage-ways.

Publications

23. ‘Healing of thermocapillary film rupture by viscous heating’ Journal of Fluid Mechanics 872, 308-326 (2019) E.Kirkinis & A.V.Andreev. The basic ideas on hydrodynamically induced destruction of malignant cells, appear in the Appendix

General odd viscosity-induced effects in viscous liquids

There is a number of striking physical manifestations associated with the presence of odd viscosity: a disk rotating in a viscous liquid experiences a normal compressive or tensile stress (Avron 1998) in addition to the shear stresses caused by the shear (even) viscosity; a swimmer experiences a torque proportional to the rate-of-change of its area (Lapa & Hughes 2014); an expanding bubble will promote an azimuthal flow on its surrounding liquid (Ganeshan & Abanov 2017), in addition to the radial flow existing in the absence of odd viscosity;

In our work we try to understand not what odd viscosity is, but what odd viscosity does, when present in an otherwise viscous liquid. We are interested in predicting new fluid-flow behavior and clarifying its consequences in mechanics. 

Publications

27. 'Null-divergence nature of the odd viscous stress for an incompressible liquid' Physical Review Fluids, 8, 014104 (2023), E.Kirkinis

25. 'Odd viscosity-induced passivation of Moffatt vortices' Journal of Fluid Mechanics, 950: A19, (2022) E. Kirkinis, J. Mason & M. Olvera de la Cruz

24. ‘Odd (or Hall) viscosity-induced stabilization of thin liquid films’ Journal of Fluid Mechanics 878, 169-189(2019) E.Kirkinis & A.V.Andreev

Nonlinear Dynamics, Pattern Formation, Boundary-Layers, Multiscale Methods

Motivated from the theory of phase transitions and critical phenomena, in my Thesis I developed a new theory to explain the underlying mechanism of the Renormalization Group (RG). This is an important method employed in statistical and high-energy physics for which the Nobel prize was awarded to Kenneth Wilson in 1982. However, in the context of applied mathematics this method remained elusive and previous attempts to rationally explain it were unsuccessful. 

I laid down its foundations based on the implicit function theorem and on a resummation  of regular asymptotic expansions 

These general closed form expansions and their amplitude equations can now be generated with symbolic computation. I also showed that the Rytov approximation (used in wave propagation in random media) forms a special case of the RG and showed the relation to Berry's phase and near-Hamiltonian systems. 

To increase the visibility of this method in applied mathematics with Hayato Chiba we organized a minisymposium in the  SIAM Conference on Applications of Dynamical Systems at Snowbird Utah (2009) and invited international experts to discuss their results (Goldenfeld, Kunihiro, Kaper and others). Recent work with Robert E. O'Malley, Jr. concentrated on applying similar methods to equations of pattern formation (Kuramoto-Sivashinsky, Swift-Hohenberg) 

Selected Publications

16. ‘Amplitude modulation for the Swift-Hohenberg and Kuramoto-Sivashinskii equations’ Journal of Mathematical Physics 55, 123510 (2014) E.Kirkinis & R.E.O’Malley, Jr.

15. ‘The Renormalization Group: A new perturbation method for the Graduate Curriculum’ SIAM Review, 54 374, (2012) E.Kirkinis

13. ‘A Combined Renormalization Group-Multiple Scale Method for Singularly Perturbed Problems’ Studies in Applied Mathematics, (4) 124 383, (2010) R.E.O’Malley, Jr. & E.Kirkinis

9. ‘Renormalization Group Interpretation of the Born and Rytov Approximations’, J.Opt.Soc.Am. A (10) 25, 2499-2508 (2008) E.Kirkinis

8. ‘The Renormalization Group and the Implicit Function Theorem for Amplitude Equations’, J. Math. Phys (7) 49, 073518 (2008) E.Kirkinis


Header Image: Museum of Science & Technology 5700 S Lake Shore Dr, Chicago, IL 60637, United States of America. Photograph by Eleftherios Kirkinis